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Ray tracing and stability analysis of parametric systems

Ray tracing and stability analysis of parametric systems. Fabrice LE BARS. Introduction Ray tracing Stability analysis of a parametric system Conclusion. > Plan. Introduction. Introduction.

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Ray tracing and stability analysis of parametric systems

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  1. Ray tracing and stability analysis of parametric systems Fabrice LE BARS

  2. Introduction • Ray tracing • Stability analysis of a parametric system • Conclusion > Plan 10/06/2009 - 2

  3. Introduction 10/06/2009 - 3

  4. Introduction • Goal : Show similarities between 2 problems apparently different : ray tracing and parametric stability analysis • Use of interval analysis 10/06/2009 - 4

  5. Ray tracing 10/06/2009 - 5

  6. Ray tracing • Description • Ray tracing, ray casting • 3D scene display • Method : build the reverse light path starting from the screen to the object 10/06/2009 - 6

  7. Ray tracing • Hypothesis • Objects are defined by implicit functions • The eye is at the origin of a coordinate space R(O,i,j,k) and the screen is at z=1 • The screen is not in the object Screen Eye Object Ray 10/06/2009 - 7

  8. Ray tracing • Problem description • A ray assiociated with the pixel satisfies 10/06/2009 - 8

  9. Ray tracing • Problem description • The point is in the object if • A pixel displays a point of the object if the associated ray intersects the object 10/06/2009 - 9

  10. Ray tracing The ray associated with intersects the object if with 10/06/2009 - 10

  11. Ray tracing • Light effects handling • Realism => illumination model • Phong : needs the distance from the eye to the object • We need to compute for each pixel : 10/06/2009 - 11

  12. Ray tracing • Computation of • If Then Moreover, if We can use a dichotomy to get 10/06/2009 - 12

  13. Ray tracing • Computation of • Interval computations are used to find • A dichotomy finds 10/06/2009 - 13

  14. Ray tracing • Parametric version • now depends on • If Then Moreover if We can use a dichotomy to get for each 10/06/2009 - 14

  15. Ray tracing • From to 10/06/2009 - 15

  16. Ray tracing 10/06/2009 - 16

  17. Stability analysis of a parametric system 10/06/2009 - 17

  18. Stability analysis of a parametric system • Stability where is retrieved from the Routh table (Routh) 10/06/2009 - 18

  19. Stability analysis of a parametric system • stability (Routh) 10/06/2009 - 19

  20. Stability analysis of a parametric system • Example : Ackermann is stable if 10/06/2009 - 20

  21. Stability analysis of a parametric system • Stability degree 10/06/2009 - 21

  22. Stability analysis of a parametric system • Similarities with ray tracing 10/06/2009 - 22

  23. Stability analysis of a parametric system 10/06/2009 - 23

  24. Conclusion 10/06/2009 - 24

  25. Conclusion • Ray tracing and stability degree drawing of a linear system are similar problems • A common algorithm based on intervals and dichotomy has been proposed 10/06/2009 - 25

  26. References • L. Jaulin. Solution globale et garantie de problèmes ensemblistes; Application à l'estimation non linéaire et à la commande robuste. PhD thesis, Université Paris XI Orsay, 1994. • S. Bazeille. Vision sous-marine monoculaire pour la reconnaissance d'objets. PhD thesis, Université de Bretagne Occidentale, 2008. • J. Flórez. Improvements in the ray tracing of implicit surfaces based on interval arithmetic. PhD thesis, Universitat de Girona, 2008. • L. Jaulin, M. Kieffer, O. Didrit et E. Walter, Applied interval analysis, Springer-Verlag, London, Great Britain, 2001. • L. Jaulin, E. Walter, O. Lévêque et D. Meizel, "Set inversion for chi-algorithms, with application to guaranteed robot localization", Math. Comput Simulation, 52, pp. 197-210, 2000. • J. Ackermann, "Does it suffice to check a subset of multilinear parameters in robustness analysis?", IEEE Transactions on Automatic Control, 37(4), pp. 487-488, 1992. • J. Ackermann, H. Hu et D. Kaesbauer, "Robustness analysis: a case study", IEEE Transactions on Automatic Control, 35(3), pp. 352-356, 1990. 10/06/2009 - 26

  27. Ray tracing 10/06/2009 - 27

  28. Ray tracing => a = d3 10/06/2009 - 28

  29. Ray tracing => b = d2 10/06/2009 - 29

  30. Ray tracing • Division in d 10/06/2009 - 30

  31. Ray tracing • Division in d 10/06/2009 - 31

  32. Ray tracing • Division in d 10/06/2009 - 32

  33. Ray tracing • Division in p 10/06/2009 - 33

  34. Ray tracing • Division in p 10/06/2009 - 34

  35. Ray tracing • Division in p 10/06/2009 - 35

  36. Ray tracing • Division in p 10/06/2009 - 36

  37. Ray tracing • Division in p 10/06/2009 - 37

  38. Ray tracing • Division in p and in d 10/06/2009 - 38

  39. Ray tracing 10/06/2009 - 39

  40. Ray tracing • Several objects display handling • We apply the previous algorithm for the function : • Indeed, we have to consider only the first object crossed by the ray 10/06/2009 - 40

  41. Ray tracing 10/06/2009 - 41

  42. Stability analysis of a parametric system • Stability degree of an invariant linear system of caracteristic polynomial P(s) : • We consider an invariant linear system parametized with a vector of parameter p : 10/06/2009 - 42

  43. Stability analysis of a parametric system • The stability degree becomes : • With the polynomial is stable if (Routh) : 10/06/2009 - 43

  44. Stability analysis of a parametric system • If we note We get • Therefore, the stability degree is : 10/06/2009 - 44

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